The sharp isomorphism on a Riemannian manifold with Riemannian metric 
is the map from the cotangent
bundle to the tangent bundle characterized
by
 |
for every covector 
and tangent vector 
. In coordinates, it sends components
to 
, where 
is the inverse metric, so
it is also called index raising. Nondegeneracy of
the metric makes the sharp isomorphism invertible;
its inverse is the flat isomorphism.
